Yoon, Dae Won; Kim, Dong-Soo; Kim, Young Ho; Lee, Jae Won Helicoidal surfaces with prescribed curvatures in \(\mathrm{Nil}_3\). (English) Zbl 1284.53058 Int. J. Math. 24, No. 14, Article ID 1350107, 11 p. (2013). Summary: We study helicoidal surfaces in the 3-dimensional Heisenberg group \(\mathrm{Nil}_3\). Also, we construct helicoidal surfaces in \(\mathrm{Nil}_3\) with prescribed Gaussian curvature or mean curvature given by smooth functions. As result, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature. Cited in 6 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C30 Differential geometry of homogeneous manifolds Keywords:Heisenberg group; helicoidal surfaces; mean curvature; Gaussian curvature PDFBibTeX XMLCite \textit{D. W. Yoon} et al., Int. J. Math. 24, No. 14, Article ID 1350107, 11 p. (2013; Zbl 1284.53058) Full Text: DOI References: [1] Arvanitoyeorgos A., JP. J. Geom. Topol. 10 pp 1– [2] DOI: 10.1007/BF01221235 · Zbl 0943.53004 · doi:10.1007/BF01221235 [3] DOI: 10.1016/S0022-247X(02)00269-X · Zbl 1022.53016 · doi:10.1016/S0022-247X(02)00269-X [4] Caddeo R., Boll. Unione. Mat. Ital. 7 pp 341– [5] Delaunay G., J. Math. Pures Appl. Ser. 6 pp 309– [6] F. Dillen, Geometry and Topology of submanifolds VIII (World Scientific Publishing, River Edge, NJ, 1996) pp. 145–147. [7] DOI: 10.1007/BF02505908 · Zbl 0965.53042 · doi:10.1007/BF02505908 [8] DOI: 10.2748/tmj/1178228808 · Zbl 0535.53002 · doi:10.2748/tmj/1178228808 [9] DOI: 10.1016/j.jmaa.2005.06.032 · Zbl 1094.53009 · doi:10.1016/j.jmaa.2005.06.032 [10] DOI: 10.2748/tmj/1178229688 · Zbl 0431.53005 · doi:10.2748/tmj/1178229688 [11] DOI: 10.1007/BF01229219 · Zbl 0940.53007 · doi:10.1007/BF01229219 [12] DOI: 10.1016/j.geomphys.2005.01.002 · Zbl 1084.53055 · doi:10.1016/j.geomphys.2005.01.002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.